Fields of definition and Belyi type theorems for curves and surfaces (Q323942)

Let \(X\subseteq\mathbb{P}^n_{\mathbb{C}}\) be a complex projective variety. For any automorphism \(\sigma\) of the field \(\mathbb{C}\), a new complex variety \(X^{\sigma}\) can be obtained by applying \(\sigma\) to the defining equations of \(X\). Let \(F\) be a countable subfield of \(\mathbb{C}\). \textit{G. González-Diez} [Q. J. Math. 57, No. 3, 339--354 (2006; Zbl 1123.14016)] that the finiteness of the set of \textit{isomorphism classes} of \(X^{\sigma},\,\sigma\in\mathrm{Aut}(\mathbb{C})\) is equivalent to the definability of \(X\) over \(F\). The paper under review revisits that paper using the language of schemes and gives a simplified proof of that theorem. The author has actually shown a refined version: The closed embedding \(X\subseteq\mathbb{P}^n_{\mathbb{C}}\) is definable over \(F\) if and only if the orbit \(\{X^{\sigma}\}_{\sigma\in\mathrm{Aut}(\mathbb{C})}\) is already finite.NEWLINENEWLINEIn the second part of the paper, Belyi's theorem for curves is reviewed and extensions to minimal surfaces are discussed. For nonruled minimal surfaces, the author gives a new proof of a theorem of \textit{G. González-Diez} [Am. J. Math. 130, No. 1, 59--74 (2008; Zbl 1158.14015)] about the definability over number fields. A new result in the same direction established in the present paper is a sufficient condition for a ruled nonrational minimal surface to be definable over number fields.